ctanhf, ctanh, ctanhl

Defined in header `<complex.h>`
float complex ctanhf( float complex z );	(1)	(since C99)
double complex ctanh( double complex z );	(2)	(since C99)
long double complex ctanhl( long double complex z );	(3)	(since C99)
Defined in header `<tgmath.h>`
#define tanh( z )	(4)	(since C99)

1-3) Computes the complex hyperbolic tangent of z.

4) Type-generic macro: If z has type long double complex, ctanhl is called. if z has type double complex, ctanh is called, if z has type float complex, ctanhf is called. If z is real or integer, then the macro invokes the corresponding real function (tanhf, tanh, tanhl). If z is imaginary, then the macro invokes the corresponding real version of the function tan, implementing the formula

tanh(iy) = i tan(y)

, and the return type is imaginary.

Parameters

z

-

complex argument

Return value

If no errors occur, complex hyperbolic tangent of z is returned

Error handling and special values

Errors are reported consistent with math_errhandling

If the implementation supports IEEE floating-point arithmetic,

ctanh(conj(z)) == conj(ctanh(z))
ctanh(-z) == -ctanh(z)
If z is +0+0i, the result is +0+0i
If z is x+∞i (for any^[1] finite x), the result is NaN+NaNi and FE_INVALID is raised
If z is x+NaN (for any^[2] finite x), the result is NaN+NaNi and FE_INVALID may be raised
If z is +∞+yi (for any finite positive y), the result is 1+0i
If z is +∞+∞i, the result is 1±0i (the sign of the imaginary part is unspecified)
If z is +∞+NaNi, the result is 1±0i (the sign of the imaginary part is unspecified)
If z is NaN+0i, the result is NaN+0i
If z is NaN+yi (for any non-zero y), the result is NaN+NaNi and FE_INVALID may be raised
If z is NaN+NaNi, the result is NaN+NaNi

↑ per DR471, this only holds for non-zero x. If z is 0+∞i, the result should be 0+NaNi
↑ per DR471, this only holds for non-zero x. If z is 0+NaNi, the result should be 0+NaNi

Notes

Mathematical definition of hyperbolic tangent is

tanh z = e z -e -z e z +e -z

Hyperbolic tangent is an analytical function on the complex plane and has no branch cuts. It is periodic with respect to the imaginary component, with period πi, and has poles of the first order along the imaginary line, at coordinates $(0, π(1/2 + n))$ . However no common floating-point representation is able to represent π/2 exactly, thus there is no value of the argument for which a pole error occurs.

Example

Run this code

#include <stdio.h>
#include <math.h>
#include <complex.h>
 
int main(void)
{
    double complex z = ctanh(1);  // behaves like real tanh along the real line
    printf("tanh(1+0i) = %f%+fi (tanh(1)=%f)\n", creal(z), cimag(z), tanh(1));
 
    double complex z2 = ctanh(I); // behaves like tangent along the imaginary line
    printf("tanh(0+1i) = %f%+fi ( tan(1)=%f)\n", creal(z2), cimag(z2), tan(1));
}

Output:

tanh(1+0i) = 0.761594+0.000000i (tanh(1)=0.761594)
tanh(0+1i) = 0.000000+1.557408i ( tan(1)=1.557408)

References

C11 standard (ISO/IEC 9899:2011):

7.3.6.6 The ctanh functions (p: 194)

7.25 Type-generic math <tgmath.h> (p: 373-375)

G.6.2.6 The ctanh functions (p: 542)

G.7 Type-generic math <tgmath.h> (p: 545)

C99 standard (ISO/IEC 9899:1999):

7.3.6.6 The ctanh functions (p: 176)

7.22 Type-generic math <tgmath.h> (p: 335-337)

G.6.2.6 The ctanh functions (p: 477)

G.7 Type-generic math <tgmath.h> (p: 480)

Compiler support
Language
Headers
Type support
Program utilities
Variadic function support
Error handling
Dynamic memory management
Strings library
Algorithms
Numerics
Date and time utilities
Input/output support
Localization support
Concurrency support (C11)
Technical Specifications
Symbol index

Common mathematical functions
Floating-point environment (C99)
Pseudo-random number generation
Complex number arithmetic (C99)
Type-generic math (C99)

csinhcsinhfcsinhl (C99)(C99)(C99)	computes the complex hyperbolic sine (function)
ccoshccoshfccoshl (C99)(C99)(C99)	computes the complex hyperbolic cosine (function)
catanhcatanhfcatanhl (C99)(C99)(C99)	computes the complex arc hyperbolic tangent (function)
tanhtanhftanhl (C99)(C99)	computes hyperbolic tangent (\({\small\tanh{x} }\) $tanh(x)$ ) (function)
C++ documentation for tanh